GROUNDS OF DISBELIEF. 381 



is a sufficient ground for disbelief, the other is so too ; and that what 

 is improbable before the fact, is therefore (not indeed in all cases, but 

 in a peculiar class of cases which I am about to specify), incredible 

 after it. , 



If, says Laplace, there are one thousand tickets in a box, and one 

 only has been drawn out; then if an eye-witness affirms that the 

 number drawji was 79, this, though the chances were 999 in 1000 

 against it, is not incredible, because the chances wei-e equally great 

 against every other number. But (he continues) if there are in the 

 box 999 black balls and only one white, and the witness affirms that 

 the white ball was drawn, this is incredible; because there was but 

 one chance in favor of white, and 999 in favor of some black ball. 



This appears to me entirely fallacious. It is evident, both from 

 general reasoning and specific experience, that the white ball will be 

 drawn out exactly as often, in any large number of trials, as the ticket 

 No. 79 will ; the two assertions, therefore, are precisely on the same 

 level in point of credibility. There is one way of putting the case 

 which, I think, must carry conviction to every one. Suppose that the 

 thousand balls are numbered, and that the white ball happens to be 

 ticketed 79. Then the drawing of the white ball, and the drawing of 

 No. 79, are the very same event; how then can the one be credible, 

 the other absolutely incredible ? A witness sees it drawn, and makes 

 his report to us : if he says that No. 79 was drawn, according to 

 Laplace, he may be believed; if he says a white ball was drawn, we 

 are bound to disbelieve him. Is this rational ] Is it not clear, on the 

 contrarr, that the only difference there could be in the credit due to 

 him would arise from moral causes, namely, from the influence which 

 (if the witness knew that there was but one white ball in a thousand) 

 might be assigned to the greater apparent wonder in the latter case "? 

 which to one kind of person would be a temptation to deceive, or to 

 take up a hasty impression, while to another, the same thing would be 

 a motive for assuring himself more positively of the fact, and would 

 therefore actually increase the credit due to his testimony. 



The mathematical reas«jning which misled Laplace into this logical 

 error, is too long to be here quoted. It is found in the section of his 

 Essai Phiiosophique stir les Probahilitcs, entitled Dc la Prohahilite des 

 Temoignages, and is founded upon a misapplication, noticed by us in 

 a former place, of his own sixth theorem of tlie doctrince of chances; a 

 theorem which he himself desci'ibes as that by which we determine 

 the probability tha^ a given effect was producecl by one or by another 

 of several causes capable of producing it. The substance of his argu- 

 ment may be briefly stated as follows : Treating the assertion of the 

 witness as the effect, lie considers as its two possible causes, the vera- 

 city or mendacity of the witness on the jjarticular occasion, that is, the 

 truth or falsity of the fact. According to the theorem, the probability 

 that the effect was produced by a particular cause, is as the antecedent 

 probability of the cause, muhiplied by the probability that the cause, 

 if it existed, would produce the given eflect. Accordingly (says 

 Laplace) in the case of the thousand tickets, the cause mendacity might 

 produce any one of 999 untrue statements, while in the case of the 

 balls, there being only two statements to make, viz., lohifc or black, 

 and one of these being true, the cause mendacity could only produce 

 one untrue statement: and consequently (the antecedent probabihty 



